If A and B are mutually exclusive events, then A and B can't both happen at the same time. Is that right?

Right.

What's the difference between that and if A and B were independent?

For starters, there is a link between mutually exclusive events- they can't both happen at once. However, there is no link between independent events- they don't effect each other at all. It might be easier to understand if you also consider non-mutually-exclusive events and dependent events.
If I draw one card from a deck, drawing an ace and drawing a king are mutually exclusive events- a single card cannot be both an ace and a king. However, drawing an ace and drawing a spade are not mutually exclusive events- a single card can be both an ace and a spade.
If I draw one card, return that card to the deck, and then draw another card, the draws are independent of each other- the sample space is the same for both draws because I returned the first card to the deck. If I draw one card, but do not return that card to the deck, and then draw another card, these events are dependent- the sample space is different since I didn't return the first card to the deck. Say I drew an ace the first time. Then there is one less card and one less ace in the deck, so the probabilities for the second draw have changed.
So mutually exclusive events are contrasted with non-mutually-exclusive events, asking whether one event excludes the other. Independent events are contrasted with dependent events, asking whether one event effects the probability of the other.